Normalized defining polynomial
\( x^{16} + 21x^{14} + 210x^{12} + 1104x^{10} + 3329x^{8} + 5124x^{6} + 4320x^{4} + 1536x^{2} + 256 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(30.65\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(420=2^{2}\cdot 3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(197,·)$, $\chi_{420}(71,·)$, $\chi_{420}(349,·)$, $\chi_{420}(223,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(169,·)$, $\chi_{420}(43,·)$, $\chi_{420}(239,·)$, $\chi_{420}(113,·)$, $\chi_{420}(307,·)$, $\chi_{420}(181,·)$, $\chi_{420}(377,·)$, $\chi_{420}(251,·)$, $\chi_{420}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{4}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{12}+\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{2}a^{6}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{16}a^{9}-\frac{1}{4}a^{7}+\frac{1}{32}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{12058248256}a^{14}+\frac{368923857}{12058248256}a^{12}+\frac{621653215}{6029124128}a^{10}+\frac{627032649}{1507281032}a^{8}-\frac{4746983423}{12058248256}a^{6}-\frac{22016030}{188410129}a^{4}-\frac{374349579}{753640516}a^{2}+\frac{86211877}{188410129}$, $\frac{1}{24116496512}a^{15}+\frac{368923857}{24116496512}a^{13}+\frac{621653215}{12058248256}a^{11}+\frac{627032649}{3014562064}a^{9}-\frac{4746983423}{24116496512}a^{7}+\frac{166394099}{376820258}a^{5}-\frac{374349579}{1507281032}a^{3}+\frac{86211877}{376820258}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{8}$, which has order $32$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{26469}{10141504}a^{14}+\frac{537853}{10141504}a^{12}+\frac{2584575}{5070752}a^{10}+\frac{3156195}{1267688}a^{8}+\frac{66896037}{10141504}a^{6}+\frac{9113817}{1267688}a^{4}+\frac{1766205}{633844}a^{2}-\frac{19642}{158461}$, $\frac{64036989}{24116496512}a^{15}+\frac{1273459029}{24116496512}a^{13}+\frac{6030101247}{12058248256}a^{11}+\frac{7242527033}{3014562064}a^{9}+\frac{155478384413}{24116496512}a^{7}+\frac{23832993713}{3014562064}a^{5}+\frac{1257699553}{188410129}a^{3}+\frac{224590592}{188410129}a$, $\frac{86479971}{24116496512}a^{15}+\frac{1808929163}{24116496512}a^{13}+\frac{8994072825}{12058248256}a^{11}+\frac{11681290437}{3014562064}a^{9}+\frac{274768798627}{24116496512}a^{7}+\frac{49159078287}{3014562064}a^{5}+\frac{17336131275}{1507281032}a^{3}+\frac{798941087}{376820258}a$, $\frac{86479971}{24116496512}a^{15}+\frac{352509}{188410129}a^{14}+\frac{1808929163}{24116496512}a^{13}+\frac{112801663}{3014562064}a^{12}+\frac{8994072825}{12058248256}a^{11}+\frac{1072632015}{3014562064}a^{10}+\frac{11681290437}{3014562064}a^{9}+\frac{2585953825}{1507281032}a^{8}+\frac{274768798627}{24116496512}a^{7}+\frac{1709907615}{376820258}a^{6}+\frac{49159078287}{3014562064}a^{5}+\frac{14891467071}{3014562064}a^{4}+\frac{17336131275}{1507281032}a^{3}+\frac{360607065}{188410129}a^{2}+\frac{798941087}{376820258}a+\frac{182351}{188410129}$, $\frac{160473763}{24116496512}a^{15}+\frac{226815}{376820258}a^{14}+\frac{3343233631}{24116496512}a^{13}+\frac{4493459}{376820258}a^{12}+\frac{16554458883}{12058248256}a^{11}+\frac{84932625}{753640516}a^{10}+\frac{1335976862}{188410129}a^{9}+\frac{405377525}{753640516}a^{8}+\frac{499789880387}{24116496512}a^{7}+\frac{268022505}{188410129}a^{6}+\frac{177828201603}{6029124128}a^{5}+\frac{291585080}{188410129}a^{4}+\frac{31461516631}{1507281032}a^{3}+\frac{451808745}{753640516}a^{2}+\frac{1448458921}{376820258}a-\frac{135818449}{188410129}$, $\frac{107882083}{24116496512}a^{15}-\frac{352509}{188410129}a^{14}+\frac{2267840671}{24116496512}a^{13}-\frac{112801663}{3014562064}a^{12}+\frac{11319913859}{12058248256}a^{11}-\frac{1072632015}{3014562064}a^{10}+\frac{1849679869}{376820258}a^{9}-\frac{2585953825}{1507281032}a^{8}+\frac{350656339267}{24116496512}a^{7}-\frac{1709907615}{376820258}a^{6}+\frac{127611699843}{6029124128}a^{5}-\frac{14891467071}{3014562064}a^{4}+\frac{21927124311}{1507281032}a^{3}-\frac{360607065}{188410129}a^{2}+\frac{507443802}{188410129}a-\frac{182351}{188410129}$, $\frac{160473763}{24116496512}a^{15}+\frac{3343233631}{24116496512}a^{13}+\frac{16554458883}{12058248256}a^{11}+\frac{1335976862}{188410129}a^{9}+\frac{499789880387}{24116496512}a^{7}+\frac{177828201603}{6029124128}a^{5}+\frac{31461516631}{1507281032}a^{3}+\frac{1448458921}{376820258}a+1$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 3121.7160225 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3121.7160225 \cdot 32}{2\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.15596069172 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_4$ (as 16T2):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_4\times C_2^2$ |
Character table for $C_4\times C_2^2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |